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C1
Modelling
29
Science
Undergraduate 3
05/31/2011

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Term
Abstract Visualization Object (AVO)
Definition

either looks like a real object or is completely abstract

based on derived information such as time, temperature, color, etc.

Term

Data fitting

Is this a good fit?

Definition
  • Occam's Razor-use as few degrees of freedom as possible
  • R2 has little meaning
  • Maximum likelihood, chi squared, and neural networks help answer this question
Term
Chi Squared
Definition

"What level of error is consistent with this linear fit model? How well is the data fit?"

Chisquared converges to a normal distribution

X~1 good fit

X<1 overfit

X>1 underfit

Term
Max Likelihood
Definition

Used in hypothesis testing.  Parameter estimation method that can limit model complexity

  1. pick a model function [f=λexp(-λt)]
  2. sum product of function
    L(t1,t2...|λ)=Πf(ti)
    L=λnexp[-λΣti]

  3.  Maximize
      dL/dλ>>>λ=n/Σin(ti)

Term

Overfitting

Why is this not advantage

Definition

Perhaps has too many degrees of freedom to fit a very finite bit of data therefore does not allow for any predictive power

 

(though, this may have a very high value of R2)

Term
What methods can be used to fit non linear data
Definition
deconvolution of gaussian fits to have a series of parabolic fits that sum together to show the overall image of the system
Term

Monte Carlo (MC) Methods

Examples

Definition

calculate the expected value of some mechanical quantity (thermodynamic quantity)

  • Buffon's Needle
  • Metropolis Algorithm
  • Microstructural evolution
  • Ising Model
Term
Metropolis Algorithm
Definition

State 1: p1, T1

State 2: p2, T2

There is a probability P* of going from state 1 to 2 (this is an ergodic perturbation)

if energy will decrease, P*=1

if energy will increase, P*=exp[-β(E2-E1)]

degrees of freedom must be considered (ie polymers can twist)

Term
Ising/Microstructural evolution
Definition

H=-εΣsisj-BΣsi

Start with perfectly ordered or perfectly random, pick a spin, flip it, recalcuate energy

 

exactly solvable and perfectly ergodic

Term

Molecular Dynamics

Types

Definition

Future state is determined by present state using force field

 

  • RDF
  • Fast Ion Conductors
Term
Principles of MD
Definition

F=f(dV,dx) gradient of velocities

F=d/dt(mv)=md2R/dt2

Verlet method: a way of advancing system by Σt

MD at constant T: Velocity scaling or Berendsen

VACF: decay of velocities based on collisions

Compute equilibrium or transport properties

Term
Velocity Scaling
Definition

An MD method of making the system at constant temperature

<KE>=1/2Σmivi2=3/2NkT

Solve for T

ΔT=T(λv)-T(v)

Term
Berendsen Method
Definition

dT/dt=1/τ(Tbath-Ti)

Integrate then plug into velocity scaling method to have a scale with only fixed variables

Term
Fast Ion Conductors
Definition
  1. Model structure
  2. diffusion model consider jumps of Q, qDE, d/2 etc. (different probability of jumps in either direction...biased)
  3. Potential energy function used
  4. RDF used to find the probaility of another particle r away
  5. Thermal expansion
  6. MSD as a function of t can be found then diffusivity derived
Term
Velocity Autocorrelation Function (VACF)
Definition

measure of how correlations in the velocities of particles in the simulation decay as a function of time.  Provides a means of investigating collision processes in molecular systems and diffusion

  • if molecules did not interact VACFv.t-flat
  • ""have weak forces-exp decay to zero
  • "" strong forces-oscillate about zero/damp
Term
What should be considered when building a model
Definition
  • target precision
  • sensitivity to precision w/ added data
  • desired outputs
  • available inputs
  • dimensional analysis
  • ease of implementation
Term
Benefits of ANN
Definition
  • Functional relationship does not have to be chosen before analysis
  • Relationship does not have to be linear
  • Fitted function can vary across input space
Term
How is complexity altered in ANN
Definition
increase the number of hidden layers/units
Term
Training v. Test Set
Definition
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mso-level-tab-stop:none; mso-level-number-position:left; text-indent:-.25in; mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"; mso-bidi-font-family:"Times New Roman";} @list l1:level9 {mso-level-number-format:roman-lower; mso-level-tab-stop:none; mso-level-number-position:right; text-indent:-9.0pt; mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"; mso-bidi-font-family:"Times New Roman";} ol {margin-bottom:0in;} ul {margin-bottom:0in;} --> training refers to the manipulation of weights and bias at random until there is a best fit to eh data obtained experimentally
  • A training set is created and compared to a test data set obtained by actually carrying out experiment
Term
Why is the metropolis algorithm not the best to use?
Definition
unphysical perterbations are valid in this model therefore it is not always representative of the true dynamics of the system
Term
Verlet method
Definition

a method of defining the time step of the system

new positions of particles are calculated using existing positions and velocities (Newton's laws)

it is ideal for the time step to be about the same as the time between collisions in a system

~10-15s

Term
Markov Process
Definition

 

 

  • present and future states are independent.  These are not varying with time, they were snapshots at different periods
  • Markov processes are ergodic and satisfy the principle of detailed balance
  • The ratio of probabilities (going from state 1 to state 2 or vis versa) needs to be Boltzmann-like (aka there is time reversal-detailed balance principle)
Term
MD v. MC
Definition
  1. (MD) Successive states are directly connected in time
  2. MD has a kinetic energy contribution to total energy while MC total energy is determined solely by potential energy
  3. MD samples from states at constant energy while MC samples from states at constant temperature

 

 

Term
Disadvantages of MD at Constant temperature
Definition
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Term
Radial distribution function (RDF)
Definition

the probability density of finding a particlei at some distance from another particlej

Related to Velocity autocorrelation function and mean square displacement

G=Σδ(r-ri)

Term
Continuous v. Impulsive MD
Definition
  • Continuous: Derive Newton's laws, advance system by some time step (use Verlet method), recalculate
    • Adv: high precision, longer time step can be used, low drift
    • Disadv: v(t) not explicitly calculated, non self-starting, must assume a value for r(t-Dt) at first duration, not fast
  • Impulsive: particles interact by discontinuous potentials ie ball bearing/hard core spheres
Term
Integrator
Definition

a finite time step acquired mathematically

 

Term

Equilibration

 

Definition

the repeated application of Markov processes. Speed of equilibration can be increased by making moves in clusters

Term
Application of ANN
Definition
crack growth is based on many parameters
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